Find the Exact Value cos(75)

Problem

cos(75^∘)

Solution

1. Identify the angle as a sum of two special angles from the unit circle.

cos(75^∘)=cos(45^∘+30^∘)

2. Apply the cosine sum formula, which states cos(A+B)=cos(A)*cos(B)−sin(A)*sin(B)

cos(45^∘+30^∘)=cos(45^∘)*cos(30^∘)−sin(45^∘)*sin(30^∘)

3. Substitute the known exact values for the trigonometric functions of 45^∘ and 30^∘

cos(45^∘)=√(,2)/2

cos(30^∘)=√(,3)/2

sin(45^∘)=√(,2)/2

sin(30^∘)=1/2

4. Multiply the fractions in each term.

√(,2)/2⋅√(,3)/2=√(,6)/4

√(,2)/2⋅1/2=√(,2)/4

5. Subtract the resulting fractions to find the final exact value.

√(,6)/4−√(,2)/4=(√(,6)−√(,2))/4

Final Answer

cos(75^∘)=(√(,6)−√(,2))/4

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